Ignition switch stuck in "on" position.

I hate to start a new thread on this but my earlier queries about replacing the ignition switch were answered. I think I'm in the weeds now, because I replaced the switch easily enough but the car still thinks the key is in the "on" position. Do need to remove the lock cylinder and do something to it, too?
This is an '01 Jetta sedan I bought from someone recently, needed "some" (ha!) exhaust work, but she didn't mention the switch was constantly open, which drains the battery. Also the immobilizer was activated, but I think I can get around that when the time comes. I already had the car running. Now I'm in the process of putting it back on the road.
Not sure how to proceed.

Recent thread with similar issue: http://forums.tdiclub.com/showthread.php?t=437157

Hm. I guess I remove the lock cylinder. I've just replaced the switch, so the problem must be at the other end...
Odd question, but a failed diode in the alternator wouldn't do this, would it?

sisyphus said:

Hm. I guess I remove the lock cylinder. I've just replaced the switch, so the problem must be at the other end...
Odd question, but a failed diode in the alternator wouldn't do this, would it?

Click to expand...

No,
cylinder lock is all mechanical part and it is broken now , that way you can not go back to off position.

I had some trouble getting the wheel off to get at the cylinder proper, so instead I removed the switch from the back of the cylinder. With the battery hooked up and no key in the cylinder the car is in "on." So I removed the switch, took a flat blade screwdriver and rotated the switch through it's positions; the car showed all the indications of various switch positions w/o starting (immobilizer issue).
I believe the cylinder itself to be sound. I think (fear, more like) the problem is further down the line.
Also, what is relay "87F diesel" for? This car doesn't have one.
On edit:
Since the door alarm chimes as soon as I hook up the new battery and the immobilizer light w/ the GP light come on and flash, along w/ the CEL, is this just an immobilizer issue, am I reading too much into this, or is there a legitimate problem? It does these things w/o a key in it. I was able to start and run the car once, before going in to fix the exhaust and some other things. But when I shut it off via the key nothing happened, it just kept running. So I shut it off by manually operating the ASV. Key turned all the way toward me (CCW), all the aforementioned lights still on.

Updated, further confounded.

So I did a continuity test between the battery leads and it showed some continuity. Not sure how serious that is. Then I hooked up the battery and while the door chime sounded and so on I started pulling fuses. Nothing changed. Moved to the three main fuses on the battery, pulled them one at a time, still no change.
Stumped.